A computational framework for investigating the stability landscape of structures is presented. A path-following algorithm is developed for mechanical systems described by sets of generalized coordinates, control/load and damage parameters undergoing quasi-static deformation. The framework makes use of an extended total potential energy functional that enables the study of structural/geometric instabilities as well as material instabilities associated with damaging processes. Instability phenomena such as distinct and compound bifurcations, secondary bifurcations (triggering mode interactions), and interactions of buckling instabilities with material damage can be studied. Beyond that, loci of critical points (critical subset paths) of geometric and material instability points can be determined. Thus, for quasi-static deformation processes, complete analyses of instability behaviours of structural members are made feasible. The capabilities of the path-following algorithm is demonstrated by studying four distinct problems: (i) the stability behaviour of a flint-arrowhead lattice structure, (ii) mode interactions in (composite) plates, (iii) the structural response of delaminated composite plates subject to in-plane compression, and (iv) interactions of buckling instabilities with material damage in composite plates.